Riemann-Roch theorems are powerful tools in algebraic geometry, connecting sheaf cohomology to geometric invariants. They extend from curves to higher-dimensional varieties, providing a way to calculate Euler characteristics and study moduli spaces.
These theorems showcase the interplay between K-theory and algebraic geometry. By expressing cohomological information in terms of Chern characters and Todd classes, they reveal deep connections between topology and algebraic structure in geometry.
Grothendieck-Riemann-Roch Theorem
Generalization of Classical Riemann-Roch Theorem
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The Grothendieck-Riemann-Roch theorem extends the classical Riemann-Roch theorem to higher dimensions and arbitrary coherent sheaves
Applies to smooth projective varieties over a field k
Expresses the Euler characteristic χ(X,F) of a coherent sheaf F on X in terms of Chern characters and Todd classes
Key Components and Definitions
The Chern character ch(F) is a ring homomorphism from the Grothendieck group K(X) of coherent sheaves on X to the Chow ring A(X)⊗Q
Encodes important information about the coherent sheaf F, such as its rank and
The Todd class td(TX) is a polynomial in the Chern classes of the tangent bundle TX
Defined using the splitting principle
Measures the "twisting" of the variety X
Grothendieck-Riemann-Roch Formula and Proof
The Grothendieck-Riemann-Roch formula states that χ(X,F)=deg(ch(F)⋅td(TX))
deg denotes the degree map from the top-dimensional component of the Chow ring to Q
The proof involves a reduction to the case of projective space using the Hirzebruch-Riemann-Roch theorem
Calculation using the splitting principle and the properties of Chern classes
Riemann-Roch Theorem for Chern Classes
Chern Character and Todd Class
For a vector bundle E, the Chern character is given by ch(E)=rank(E)+c1(E)+21(c1(E)2−2c2(E))+⋯
For a coherent sheaf F, ch(F) is defined using a locally free resolution of F
The Todd class of the tangent bundle TX is given by td(TX)=1+21c1(TX)+121(c1(TX)2+c2(TX))+⋯
Interpretation and Special Cases
The Riemann-Roch formula χ(X,F)=deg(ch(F)⋅td(TX)) can be seen as a "correction" to the naive Euler characteristic
Takes into account the Chern classes of F and the geometry of X
For a L, the formula simplifies to χ(X,L)=deg(ch(L)⋅td(TX))=21deg(c1(L)dim(X)+lower order terms)
Recovers the classical Riemann-Roch theorem
Euler Characteristic Calculations
Definition and Properties
The Euler characteristic χ(X,F) is defined as the alternating sum of the dimensions of the cohomology groups Hi(X,F)
Can be computed using the Riemann-Roch theorem in various settings
Examples and Applications
For a line bundle L on a smooth projective curve X, χ(X,L)=deg(L)+1−g, where g is the of X
For a vector bundle E on a smooth projective surface X, χ(X,E)=21deg(c1(E)2−2c2(E))+121deg(c1(TX)c1(E))+121deg(c1(TX)2+c2(TX))
In higher dimensions, the Euler characteristic can be computed using the Hirzebruch-Riemann-Roch theorem or the Grothendieck-Riemann-Roch theorem
The Riemann-Roch theorem is used to study the geometry of moduli spaces of sheaves on varieties
Computes expected dimensions and constructs virtual fundamental classes
Riemann-Roch Generalization for Singular Varieties
Higher K-Theory and Chern Character
The Grothendieck-Riemann-Roch theorem generalizes to singular varieties using intersection theory and the Chern character in higher K-theory
For a possibly singular variety X, the Grothendieck group K(X) is replaced by the higher K-theory group K0(X)
K0(X) is the Grothendieck group of perfect complexes on X
The Chern character ch:K0(X)→A(X)⊗Q is defined using the Atiyah-Hirzebruch spectral sequence
Virtual Tangent Bundle and Generalizations
In the presence of singularities, the Todd class is replaced by the Todd class of the virtual tangent bundle
Defined using the cotangent complex
The Riemann-Roch theorem can be generalized to higher K-theory groups Ki(X) for i>0
Uses the Chern character in higher K-theory and the Adams operations
These generalizations have applications in the study of characteristic classes of singular varieties and in the construction of virtual fundamental classes in enumerative geometry
Key Terms to Review (16)
Adelic Riemann-Roch theorem: The adelic Riemann-Roch theorem is a powerful result in algebraic geometry that generalizes classical Riemann-Roch theory to the setting of arithmetic surfaces and schemes. This theorem connects the geometry of a variety with its arithmetic properties, using adelic methods to provide a way to compute dimensions of certain spaces of sections of line bundles over varieties, taking into account both local and global aspects.
Bernhard Riemann: Bernhard Riemann was a German mathematician who made significant contributions to analysis, differential geometry, and number theory in the 19th century. His work laid the foundation for many modern mathematical concepts, particularly through his formulation of the Riemann-Roch theorem, which is essential in algebraic geometry for understanding the relationship between algebraic curves and their divisors.
Chern classes: Chern classes are a set of characteristic classes associated with complex vector bundles, providing vital topological invariants that help classify vector bundles over a manifold. They connect deeply with various fields such as geometry, topology, and algebraic geometry, allowing us to analyze vector bundles through their topological properties.
Degree of a divisor: The degree of a divisor is a fundamental concept in algebraic geometry that quantifies the number of points at which a divisor intersects a given divisor, counted with multiplicity. This degree provides crucial information about the behavior of meromorphic functions on algebraic curves and is closely linked to the Riemann-Roch theorem, which connects the geometry of divisors with linear systems and the dimensions of space of meromorphic functions.
Dimension of global sections: The dimension of global sections refers to the size or dimensionality of the space formed by global sections of a sheaf on a given space. This concept plays a crucial role in understanding the relationships between sheaves, cohomology, and algebraic geometry, especially when applying the Riemann-Roch theorem to analyze the properties of line bundles and divisors.
Divisor: A divisor is a formal way to assign a formal sum of points to a divisor on an algebraic curve or variety, indicating how a function behaves at different points. It plays a critical role in the study of algebraic geometry, particularly in the context of Riemann-Roch theorems, which connect divisors to important properties like linear equivalence and the existence of certain types of functions.
Effective Divisor: An effective divisor is a formal sum of irreducible subvarieties of a given algebraic variety, with non-negative integer coefficients. In algebraic geometry, effective divisors correspond to points or subvarieties that can be associated with certain functions or sections, providing a way to understand the relationship between algebraic curves and their geometric properties.
Generalized Riemann-Roch theorem: The generalized Riemann-Roch theorem is a central result in algebraic geometry that extends the classical Riemann-Roch theorem to a broader setting, particularly for coherent sheaves on algebraic varieties. It establishes a deep connection between the dimensions of spaces of sections of sheaves, the characteristics of the underlying variety, and the divisor class associated with the sheaf. This theorem plays a crucial role in understanding the properties of line bundles and their sections, facilitating various applications in intersection theory and the study of algebraic curves.
Genus: In the context of algebraic geometry and the Riemann-Roch theorems, the genus refers to a topological invariant that classifies algebraic curves based on their shape and complexity. The genus is essential for understanding the properties of curves, including their behavior under various mappings and the dimensions of spaces of meromorphic functions on them.
Gunnar K. R. H. E. L. A. Rosenlicht: Gunnar K. R. H. E. L. A. Rosenlicht is a prominent mathematician known for his contributions to the Riemann-Roch theorem in algebraic geometry. His work has provided significant insights into the connections between divisors on algebraic curves and the dimensions of associated vector spaces, influencing both theoretical and applied mathematics.
Line Bundle: A line bundle is a mathematical construct in algebraic geometry that generalizes the notion of a product of a space with the complex numbers, forming a vector space of dimension one. Line bundles are essential for studying various geometric properties, such as divisors, sheaves, and cohomology, and play a critical role in the formulation of the Riemann-Roch theorem, which relates the geometry of algebraic curves to their function spaces.
Non-singular points: Non-singular points are points on a variety where the local structure behaves nicely, meaning the tangent space has the expected dimension and there are no singularities or irregularities present. These points are essential in algebraic geometry as they allow for well-defined properties and behaviors of functions and curves, facilitating the application of tools like the Riemann-Roch theorem.
Riemann-Roch Function: The Riemann-Roch function is a crucial concept in algebraic geometry that provides a powerful tool for studying the properties of divisors on algebraic curves. It helps in calculating dimensions of spaces of meromorphic functions and forms with prescribed poles and zeros. This function establishes a relationship between the geometry of curves and algebraic properties, providing insights into the interplay between divisors and their associated linear systems.
Riemann-Roch Space: Riemann-Roch space refers to a specific vector space of meromorphic functions or differentials on a compact Riemann surface that satisfies certain conditions related to divisor theory. This space is crucial in algebraic geometry as it provides insight into the relationship between the geometry of the surface and its function theory, particularly in understanding how divisors correspond to linear systems of functions.
Sheaf Theory: Sheaf theory is a mathematical framework that allows for the systematic study of local data in a coherent way across different spaces, particularly useful in topology and algebraic geometry. By associating algebraic structures (like sets, groups, or rings) to open sets of a topological space, sheaves enable mathematicians to tackle problems involving local-to-global principles, making them essential for understanding various properties of spaces and functions defined on them.
Smooth curve: A smooth curve is a continuous curve that has continuous derivatives up to a required order, allowing for no sharp corners or cusps. In the context of algebraic geometry, smooth curves are crucial as they enable the application of the Riemann-Roch theorems, which provide important insights into the relationship between algebraic curves and their function theory.